Subgroups — #44402603

Order 29056.

604 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20300247	C₂	1	2	yes	maximal
2	#28408931	C₂	1	2	yes	maximal, complemented
3	#28408918	C₂	1	4	yes	maximal, complemented
4	#28408895	C₂	1	2	yes	maximal
5	#28408892	C₂	1	1	yes	maximal, characteristic
6	C₄⋊C₄×Dic₂₂₇	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊Dic₂₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄⋊C₄×C₂₂₇	C₂	1	1	yes	maximal, characteristic
9	#20300247	—	2	8	no	—
10	C₂×C₄⋊Dic₂₂₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₂₂₇	C₂²	1	4	yes	—
12	C₂×C₂²⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊Dic₂₂₇	C₂²	1	2	yes	—
14	C₂×C₂²⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₂₂₇	C₂²	1	2	yes	—
16	C₂×C₄×Dic₂₂₇	C₂²	1	4	yes	—
17	C₂₂₇⋊#95	C₂²	1	2	yes	—
18	C₂₂₇⋊#95	C₂²	1	4	yes	—
19	C₂₂₇⋊#95	C₂²	1	2	yes	—
20	C₂₂₇⋊#95	C₂²	1	4	yes	—
21	C₂³×C₄×C₂₂₇	C₂²	1	1	yes	characteristic
22	C₂³×C₄×C₂₂₇	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₂₂₇	C₄	1	8	yes	complemented
24	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	2	yes	—
26	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
27	C₂²×Dic₂₂₇	C₂³	1	2	yes	—
28	C₂²×Dic₂₂₇	C₂³	1	4	yes	—
29	C₂²×Dic₂₂₇	C₂³	1	2	yes	—
30	C₄⋊Dic₂₂₇	—	2	8	no	—
31	C₄⋊Dic₂₂₇	—	2	4	no	—
32	C₂²⋊Dic₂₂₇	—	2	4	no	—
33	C₂²⋊Dic₂₂₇	—	2	2	no	—
34	C₂²⋊Dic₂₂₇	D₄	1	4	yes	—
35	C₄×Dic₂₂₇	—	2	8	no	—
36	C₄×Dic₂₂₇	—	2	4	no	—
37	C₂⁴×C₂₂₇	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₂₂₇	—	2	4	no	—
39	C₂²×C₄×C₂₂₇	—	2	4	no	—
40	C₂²×C₄×C₂₂₇	—	2	4	no	—
41	C₂²×C₄×C₂₂₇	C₂×C₄	1	8	yes	—
42	C₂²×C₄×C₂₂₇	C₂×C₄	1	4	yes	—
43	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₂₂₇	C₂×C₄	1	8	yes	—
48	C₄⋊C₄×C₂₂₇	C₂×C₄	1	4	yes	—
49	C₄⋊C₄×C₂₂₇	C₂×C₄	1	4	yes	—
50	C₂×Dic₂₂₇	—	2	2	no	—
51	C₂×Dic₂₂₇	—	2	2	no	—
52	C₂×Dic₂₂₇	—	2	4	no	—
53	C₂×Dic₂₂₇	—	2	2	no	—
54	C₂×Dic₂₂₇	—	2	2	no	—
55	C₂×Dic₂₂₇	—	2	4	no	—
56	C₂×Dic₂₂₇	—	2	4	no	—
57	C₂×Dic₂₂₇	—	2	2	no	—
58	C₂×Dic₂₂₇	C₂×D₄	1	4	yes	—
59	C₂³×C₂₂₇	—	2	2	no	—
60	C₂³×C₂₂₇	—	2	2	no	—
61	C₂³×C₂₂₇	C₂²×C₄	1	2	yes	—
62	C₂³×C₂₂₇	C₂×D₄	1	2	yes	—
63	C₂³×C₂₂₇	#41	1	2	yes	—
64	C₂³×C₂₂₇	C₂⁴	1	1	yes	characteristic
65	C₂×C₄×C₂₂₇	—	2	16	no	—
66	C₂×C₄×C₂₂₇	—	2	2	no	—
67	C₂×C₄×C₂₂₇	—	2	2	no	—
68	C₂×C₄×C₂₂₇	—	2	2	no	—
69	C₂×C₄×C₂₂₇	—	2	2	no	—
70	C₂×C₄×C₂₂₇	—	2	2	no	—
71	C₂×C₄×C₂₂₇	—	2	8	no	—
72	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₂₂₇	C₂²×C₄	1	4	yes	—
74	C₂×C₄×C₂₂₇	C₂²×C₄	1	4	yes	—
75	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₂₂₇	#41	1	4	yes	—
77	C₂²×C₂₂₇	—	2	2	no	—
78	C₂²×C₂₂₇	—	2	2	no	—
79	C₂²×C₂₂₇	—	2	4	no	—
80	C₂²×C₂₂₇	—	2	4	no	—
81	C₂²×C₂₂₇	C₄×D₄	1	4	yes	—
82	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
83	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
84	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
85	C₂²×C₂₂₇	C₂×#41	1	2	yes	—
86	C₂²×C₂₂₇	2₋¹⁺⁴	1	2	yes	—
87	C₄×C₂₂₇	—	2	2	no	—
88	C₄×C₂₂₇	—	2	4	no	—
89	C₄×C₂₂₇	—	2	4	no	—
90	C₄×C₂₂₇	#117	1	4	yes	complemented
91	Dic₂₂₇	—	2	4	no	—
92	Dic₂₂₇	—	4	4	no	—
93	Dic₂₂₇	—	4	2	no	—
94	C₂×C₂₂₇	—	2	4	no	—
95	C₂×C₂₂₇	C₂×#117	1	1	yes	characteristic
96	C₂×C₂₂₇	C₂×C₄×D₄	1	1	yes	characteristic
97	C₂×C₂₂₇	#519	1	1	yes	characteristic
98	C₂×C₂₂₇	#547	1	2	yes	—
99	C₂×C₂₂₇	#552	1	2	yes	—
100	C₂₂₇	#2344	1	1	yes	characteristic, complemented, Sylow
101	#2344	—	227	1	no	maximal, Sylow
102	C₂×#117	—	227	2	no	—
103	C₂²×C₄⋊C₄	D₂₂₇	1	1	yes	characteristic
104	#389	—	227	2	no	—
105	#387	—	227	4	no	—
106	#381	—	227	2	no	—
107	#380	—	227	1	no	—
108	C₄×C₄⋊C₄	—	227	2	no	—
109	C₄×C₂²⋊C₄	—	227	1	no	—
110	C₂³×C₄	C₂×D₂₂₇	1	2	yes	—
111	C₂³×C₄	C₂×D₂₂₇	1	1	yes	characteristic
112	#117	—	454	8	no	—
113	C₂×C₄⋊C₄	—	227	4	no	—
114	C₂×C₄⋊C₄	—	227	2	no	—
115	C₂×C₄⋊C₄	Dic₂₂₇	1	8	yes	complemented
116	C₂×C₄⋊C₄	C₂×D₂₂₇	1	1	yes	characteristic
117	C₂×C₄⋊C₄	C₂×D₂₂₇	1	2	yes	—
118	C₂×C₄⋊C₄	C₂×D₂₂₇	1	1	yes	characteristic
119	C₂×C₂²⋊C₄	—	227	2	no	—
120	C₂×C₂²⋊C₄	—	227	1	no	—
121	C₂×C₂²⋊C₄	—	227	1	no	—
122	C₂×C₄²	—	227	4	no	—
123	C₂×C₄²	—	227	2	no	—
124	#95	—	227	2	no	—
125	#95	—	227	4	no	—
126	#95	—	227	2	no	—
127	#95	—	227	4	no	—
128	C₂⁴	C₂²×D₂₂₇	1	1	yes	characteristic
129	C₂²×C₄	—	2	4	no	—
130	C₂²×C₄	—	2	4	no	—
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	—	227	2	no	—
133	C₂²×C₄	—	227	4	no	—
134	C₂²×C₄	—	227	2	no	—
135	C₂²×C₄	C₂×Dic₂₂₇	1	8	yes	—
136	C₂²×C₄	C₂×Dic₂₂₇	1	4	yes	—
137	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
138	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
139	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
140	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
141	C₄⋊C₄	—	454	8	no	—
142	C₄⋊C₄	—	454	4	no	—
143	C₄⋊C₄	C₂×Dic₂₂₇	1	8	yes	—
144	C₄⋊C₄	C₂×Dic₂₂₇	1	4	yes	—
145	C₄⋊C₄	C₂×Dic₂₂₇	1	4	yes	—
146	C₂²⋊C₄	—	454	4	no	—
147	C₂²⋊C₄	—	454	2	no	—
148	C₂²⋊C₄	—	227	4	no	—
149	C₄²	—	454	8	no	—
150	C₄²	—	454	4	no	—
151	C₂³	—	2	2	no	—
152	C₂³	—	2	2	no	—
153	C₂³	C₂²×Dic₂₂₇	1	2	yes	—
154	C₂³	D₄×D₂₂₇	1	2	yes	—
155	C₂³	#16813883	1	2	yes	—
156	C₂³	C₂³×D₂₂₇	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	16	no	—
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	454	4	no	—
165	C₂×C₄	—	454	2	no	—
166	C₂×C₄	—	454	2	no	—
167	C₂×C₄	—	454	4	no	—
168	C₂×C₄	—	454	4	no	—
169	C₂×C₄	—	454	2	no	—
170	C₂×C₄	—	454	2	no	—
171	C₂×C₄	—	454	2	no	—
172	C₂×C₄	—	227	4	no	—
173	C₂×C₄	C₂²×Dic₂₂₇	1	2	yes	—
174	C₂×C₄	C₂²×Dic₂₂₇	1	4	yes	—
175	C₂×C₄	C₂²×Dic₂₂₇	1	4	yes	—
176	C₂×C₄	C₂²×Dic₂₂₇	1	2	yes	—
177	C₂×C₄	#16813884	1	4	yes	—
178	C₂²	—	2	4	no	—
179	C₂²	—	2	4	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	2	no	—
182	C₂²	D₄×Dic₂₂₇	1	4	yes	—
183	C₂²	C₂³×Dic₂₂₇	1	1	yes	characteristic
184	C₂²	C₂×D₄×D₂₂₇	1	1	yes	characteristic
185	C₂²	C₂×#16813883	1	1	yes	characteristic
186	C₂²	C₂×#16813884	1	2	yes	—
187	C₂²	#20300336	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	4	no	—
191	C₄	—	908	4	no	—
192	C₄	—	908	2	no	—
193	C₄	—	454	4	no	—
194	C₄	#20300247	1	4	yes	complemented
195	C₂	—	2	4	no	—
196	C₂	C₂×#20300247	1	1	yes	characteristic
197	C₂	C₂×D₄×Dic₂₂₇	1	1	yes	characteristic
198	C₂	#28409510	1	1	yes	characteristic
199	C₂	#28409683	1	2	yes	—
200	C₂	#28409717	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₂₇
commutator	C₂²×C₂₂₇	C₂³×C₄
Fitting	C₂²×C₄⋊C₄×C₂₂₇	C₂
socle, cosocle	C₂³×C₂₂₇	C₂⁴
prime radical	C₂⁴×C₂₂₇	C₂³
hypercenter	C₂²×C₄⋊C₄	D₂₂₇
hypocenter	C₂₂₇	#2344
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #28408931	C₂	semidirect
3. #28408918	C₂	semidirect
6. C₄⋊C₄×Dic₂₂₇	C₂	semidirect
23. C₂×C₄⋊C₄×C₂₂₇	C₄	semidirect
90. C₄×C₂₂₇	#117	semidirect
100. C₂₂₇	#2344	semidirect
115. C₂×C₄⋊C₄	Dic₂₂₇	semidirect
194. C₄	#20300247	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#20300247	2
#28408931	2
#28408918	2
#28408895	2
#28408892	2
C₄⋊C₄×Dic₂₂₇	2
C₄×C₂²⋊Dic₂₂₇	2
C₂²×C₄⋊C₄×C₂₂₇	2
#2344	227